驳斥Fejes Tóth关于曲面网格精度的说法

IF 0.4 4区数学 Q4 MATHEMATICS

Studia Scientiarum Mathematicarum Hungarica Pub Date : 2020-06-01 DOI:10.1556/012.2020.57.2.1454

G. Vegter, M. Wintraecken

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引用次数: 0

摘要

Fejes Tóth[3]研究了在三维空间中光滑表面的近似，通过在表面上给定顶点数量的分段平坦三角形网格，这些顶点相对于Hausdorff距离是最优的。他证明了如果曲面是凸的，这个Hausdorff距离与近似网格的顶点数成反比。他还声称这个豪斯多夫距离与一个特定的非凸曲面的顶点数的平方成反比，即一个由两个全等圆包围的单面旋转双曲面。我们反驳了这一说法，并证明了Hausdorff距离的渐近行为是线性的，这与凸曲面的渐近行为是相同的。

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Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes

Fejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

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来源期刊

Studia Scientiarum Mathematicarum Hungarica 数学-数学

CiteScore

1.30

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original research papers on various fields of mathematics, e.g., algebra, algebraic geometry, analysis, combinatorics, dynamical systems, geometry, mathematical logic, mathematical statistics, number theory, probability theory, set theory, statistical physics and topology.